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Theorem bj-inf2vnlem3 7386
Description: Lemma for bj-inf2vn 7388. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-inf2vnlem3.bd1 BOUNDED A
bj-inf2vnlem3.bd2 BOUNDED 𝑍
Assertion
Ref Expression
bj-inf2vnlem3 (x A (x = ∅ y A x = suc y) → (Ind 𝑍A𝑍))
Distinct variable groups:   x,y,A   x,𝑍,y

Proof of Theorem bj-inf2vnlem3
Dummy variables z 𝑡 u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-inf2vnlem2 7385 . . 3 (x A (x = ∅ y A x = suc y) → (Ind 𝑍u(𝑡 u (𝑡 A𝑡 𝑍) → (u Au 𝑍))))
2 bj-inf2vnlem3.bd1 . . . . . 6 BOUNDED A
32bdeli 7265 . . . . 5 BOUNDED z A
4 bj-inf2vnlem3.bd2 . . . . . 6 BOUNDED 𝑍
54bdeli 7265 . . . . 5 BOUNDED z 𝑍
63, 5ax-bdim 7233 . . . 4 BOUNDED (z Az 𝑍)
7 nfv 1398 . . . 4 z(𝑡 A𝑡 𝑍)
8 nfv 1398 . . . 4 z(u Au 𝑍)
9 nfv 1398 . . . 4 u(z Az 𝑍)
10 nfv 1398 . . . 4 u(𝑡 A𝑡 𝑍)
11 eleq1 2078 . . . . . 6 (z = 𝑡 → (z A𝑡 A))
12 eleq1 2078 . . . . . 6 (z = 𝑡 → (z 𝑍𝑡 𝑍))
1311, 12imbi12d 223 . . . . 5 (z = 𝑡 → ((z Az 𝑍) ↔ (𝑡 A𝑡 𝑍)))
1413biimpd 132 . . . 4 (z = 𝑡 → ((z Az 𝑍) → (𝑡 A𝑡 𝑍)))
15 eleq1 2078 . . . . . 6 (z = u → (z Au A))
16 eleq1 2078 . . . . . 6 (z = u → (z 𝑍u 𝑍))
1715, 16imbi12d 223 . . . . 5 (z = u → ((z Az 𝑍) ↔ (u Au 𝑍)))
1817biimprd 147 . . . 4 (z = u → ((u Au 𝑍) → (z Az 𝑍)))
196, 7, 8, 9, 10, 14, 18bdsetindis 7383 . . 3 (u(𝑡 u (𝑡 A𝑡 𝑍) → (u Au 𝑍)) → z(z Az 𝑍))
201, 19syl6 29 . 2 (x A (x = ∅ y A x = suc y) → (Ind 𝑍z(z Az 𝑍)))
21 dfss2 2907 . 2 (A𝑍z(z Az 𝑍))
2220, 21syl6ibr 151 1 (x A (x = ∅ y A x = suc y) → (Ind 𝑍A𝑍))
Colors of variables: wff set class
Syntax hints:  wi 4   wo 616  wal 1224   = wceq 1226   wcel 1370  wral 2280  wrex 2281  wss 2890  c0 3197  suc csuc 4047  BOUNDED wbdc 7259  Ind wind 7345
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-bdim 7233  ax-bdsetind 7382
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-sn 3352  df-suc 4053  df-bdc 7260  df-bj-ind 7346
This theorem is referenced by:  bj-inf2vn  7388
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